2020/12/28 by Yuri A. Kordyukov, Kordyukov, Yuri A.
Mathematics · Medicine · #Advanced Algebra and Geometry #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Geometry and complex manifolds #Mathematical Physics (math-ph) #Medical Imaging Techniques and Applications #Spectral Theory (math.SP)
paper · pdf · doi:10.48550/arxiv.2012.14198
openalex publication_date 2020/12/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we construct a family of Berezin-Toeplitz type quantizations of a compact symplectic manifold. For this, we choose a Riemannian metric on the manifold such that the associated Bochner Laplacian has the same local model at each point (this is slightly more general than in almost-Kähler quantization). Then the spectrum of the Bochner Laplacian on high tensor powers Lp of the prequantum line bundle L asymptotically splits into clusters of size \mathcal O(p3/4) around the points pΛ, where Λ is an eigenvalue of the model operator (which can be naturally called a Landau level). We develop the Toeplitz operator calculus with the quantum space, which is the eigenspace of the Bochner Laplacian corresponding to the eigebvalues frrom the cluster. We show that it provides a Berezin-Toeplitz quantization. If the cluster corresponds to a Landau level of multiplicity one, we obtain an algebra of Toeplitz operators and a formal star-product. For the lowest Landau level, it recovers the almost Kähler quantization.